Topologically protected entanglement switching around exceptional points

The robust operation of quantum entanglement states is crucial for applications in quantum information, computing, and communications1–3. However, it has always been a great challenge to complete such a task because of decoherence and disorder. Here, we propose theoretically and demonstrate experimentally an effective scheme to realize robust operation of quantum entanglement states by designing quadruple degeneracy exceptional points. By encircling the exceptional points on two overlapping Riemann energy surfaces, we have realized a chiral switch for entangled states with high fidelity. Owing to the topological protection conferred by the Riemann surface structure, this switching of chirality exhibits strong robustness against perturbations in the encircling path. Furthermore, we have experimentally validated such a scheme on a quantum walk platform. Our work opens up a new way for the application of non-Hermitian physics in the field of quantum information.

...  Taking the parameters 1  and  as variables, by solving the eigen-equation Figure 1b shows the real part of the quasienergy as a function of 1  and  .Four energy surfaces are divided into two groups where each group contains two degenerated Riemann energy surfaces.An isolated EP (green sphere in Fig. 1b) exists at the branch point of these surfaces.
The red or blue color indicates that the imaginary part of quasi-energy   is positive or negative, respectively.It is found that the evolution process described in Fig. 1a   The origin for such asymmetric conversion is uncovered below.For the input state 1  , its real part of the energy is less than 0, which is described by the white asterisk in Fig. 1b.So when starting from 1  and encircling the EP clockwise, the state experiences the evolution path on the red Riemann surface, i.e. experiencing the gain mode where the imaginary part of the quasienergy is positive.In this case, the input states adiabatically evolve on the Riemann surface, and change to 2  after one cycle of parameter changes, as shown in Fig. 2e.While, when the input state changes to 2  (its real part of the energy greater than 0, labeled as yellow asterisk in Fig. 1b), the evolution path encircling the EP clockwise is on the blue Riemann surface at the initial stage, i.e. the loss mode where the imaginary part of the quasienergy is negative.In this case, the tiny non-adiabatic coupling between the loss and gain modes of the non-Hermitian system induces non-adiabatic transitions, which breaks the adiabaticity.It results in the transition from the blue Riemann surface to the red one during the evolution, eventually return to itself after one cycle, as shown in Fig. 2f  can also be realized through encircling the EP.The origin for such asymmetric conversion is also similar to that for 1  and 2  .
The study above demonstrates that encircling the EP enables asymmetric conversion between the four entangled states, i.e., realize a chirality switch for entangled states.The output entangled state in the conversion is determined by the direction of circling the EP, and the conversion efficiency is very high.This phenomenon can be attributed to the consistency between the four eigenstates ( 1, 2,3, 4) Furthermore, it is emphasized that the above manipulation processes for the entanglement states are topologically protected due to topological properties of EP.And more importantly, these phenomena can all be experimentally demonstrated.In the following, we discuss the experimental realization of the above theoretical scheme by constructing the non-Hermitian QW platform, and demonstrate the robustness of this chiral switch.

Experimental realization of topological entanglement switching. The constructed non-
Hermitian QW platform is shown in Fig. 3, which contains three parts: state preparation (source), evolution process, and measurement.This corresponds to the theoretical scheme in Fig. 1a.In the state preparation, we first use 400 nm picosecond laser pulses to pump a 3 mm thick  -BaB2O4 (BBO) crystals, generating photon pairs at 800 nm through type-I parametric down conversion.These photon pairs are sent through interference filters to enhance their indistinguishability and coupled into single-mode fibers.The quantum states 0 and 1 of the two particles are encoded in the horizontal ( H ) and vertical ( V ) polarization states of the two photons, respectively.In the experiment, we choose the four maximally entangled Bell  After preparation, the photons are then sent into the multi-step QW     Compared with the previous theoretical design, the intermediate  and  , the loops 1 and 2 described theoretically in Fig. 1c are implemented experimentally.
For the last operation After the two photons undergo the above evolution process, the output state is obtained through two-photon quantum state tomography.By using an apparatus consisting of QWPs, HWPs and polarizers, 16 measurement bases are constructed to perform projective measurements on the output state.With these projective measurement results, the quantum state tomography is completed and the density matrix of the output state is reconstructed.th ex S  between theoretical and experimental results.
The experimental results for the output states are shown in Fig. 4. In the experiment, a total number of QW steps 8 N  is taken.In fact, the theoretical results shown in Fig. 2 exhibit the case with 100-steps QW, which the parameters are unequally spaced.It is very difficult to realize experimentally with so many QW steps due to loss.Fortunately, it is found that good results can be obtained using fewer QW steps when the parameters 1  and  are unequally spaced.This is because it is uneven for the matching degree between the output state and input state for each step evolution along Loops.The calculated results with In the experiments, the fidelity      is the theoretical density matrix.It can be seen that the similarity for all cases is greater than 92%, indicating the excellent agreement between experiment and theory.This means that we have successfully experimentally demonstrated the chiral switch for the four Bell states.The inevitable loss of photon leads to the resource of error, and the related analysis in the experiment has been provided in S4 of Supplementary Materials.
In addition, when choosing the red Loop 2 in Fig. 1c, the experimental results show that the chiral behavior disappears, which are also identical with theoretical results, see S5 of Supplementary Materials.In order to verify the robustness of this switching behavior, the disorder is introduced into the encircled path, and to observe the variations in the output entangled states.In the experiment, the is realized by adding small random angular deviations the rotation angles of the waveplates, i.e., the parameters of path become 11   and   , where the disorder strengths   and   are uniformly random chosen within the interval   0.025,0.025 rad.
Here, 1  and  take the same values as those in the Fig.This means that the phenomena we have revealed are easier to be implemented in various real systems, which is very beneficial for future quantum information, computing, and communications.

The details of pre-and post-control operators.
In our discussion, the operators for the nth QW can be expressed as . The terms The elements At the nth step, two photons undergo the evolution as n IM  , which can be described as 0 0 0 0 00 00 00 00 For the above operator n IM  , it is obviously that the eigenstates are not the Bell states.
Based on the studies about quantum state control encircling the EP, the efficient control among Bell states requires the eigenstates of system to be nearly Bell states.Therefore, to realize the efficient control of Bell states, we add the pre-and post-control operators n C and Here, the column vectors in the matrix A are composed of the eigenstates of n U .For the operator n IM  , we can also have the eigen-equation as, where the column vectors in the matrix B are composed of the eigenstates of n IM  .By combing Eq. ( 5) and ( 6), we can obtain the relation as, The operator n C has the form as The operator n U corresponds to the one step evolution for two photons, and can be expressed 2. The construction of QW with Bell states as its eigenstates.
By solving the eigen-equation, the eigenstates of n U can be obtained as   .(10) In our study, the parameters for the starting point of system are chosen as    

in the design of experiment
When the evolution encircles the EP, the output state is obtained as: By replacing   , the above equation changes to: In our study, the state changes slowly on the Riemann energy surface with  and ( For our discussions in the main text, we also numerically calculate the Eq. ( 13) and find it is always satisfied.In this way, the evolution shown in Eq. ( 12) can be simplified as: The Eq. ( 14) means that we can achieve the circle of the EP following this simple evolution.
In our experiment, the input states undergo the optical elements consisting of the evolution as Eq. ( 14), and change to different output states depending on the circle of the EP clockwise and counter-clockwise.Next, we show how to realize the operators in the optical platform.

Realizations of operators in the optical platform.
In the experiment, the combination of two QWPs and one HWP can realize any unitary operation of single polarization bits.Here, we design a specific combination of waveplates to achieve a more concise form of this evolution.
The rotation operator R can be achieved by the combination of two half-wave pieces with angles 0 and  respectively.The symbol '  ' represents the actions of operators from right to left.

ii.
Conditional phase shift operator S .
Therefore, the operator S can be realized by combining two QWPs with an angle of 2  and one HWP with an angle of 2 k   . iii.
The equivalent gain-loss operator L and ' L .
The polarization-dependent loss operators In order to realize such a PPBS with a special polarization transmittance, we add two HWPs with a rotation angle of 4  before and after injecting to the PPBS.So the operator ' L can be experimentally realized by a sandwich-type HWP-PPBS-HWP combination.

iv. The symmetry breaking operator
. (17) The operator  can be realized by combining two QWPs with an angle of 0 and one HWP with an angle of  .We have provided the realization of C  can be obtained with Eqs. ( 5)- (7).
When the input states are Bell states As shown by Eqs. ( 19)- (20), after going through the operator 1   1 C  , the Bell states change to new product states.These product states can be realized through adding QWP and HWP to the two photons generated at the  -BaB2O4 (BBO) crystals.

1 iCR 1 G 1 IM  , they are both acted upon by the operator 1 1 C
 .The detailed expressions and derivations of these control operators are provided in Section 1 of Methods.Taking the first evolution step operator , the single step evolution process can be divided into three stages.First, both particles are acted upon by the operator 1 C .Next, the red particle enters the identity matrix module I , while the gray particle enters the QW module 1  acts on the gray particle, it can make the output state a linear superposition state related to 0 and 1 phase shift operator, which adds a phase shift of ik e for state 0 , and the opposite phase shift of ik e  for state 1 .The gain- , where  is the gain-loss strength.Under the action of G , the wave function with the state 0 ( 1 ) is amplified (reduced).The effect of  is the opposite of G .The symmetry breaking operator is  .Afterwards, each evolution step operator   i U i N  acting on the two particles follows a similar three-stage process, with the difference being that the parameters 1  and  in the evolution change.

Figure 1 .
Figure 1.The realization of the dynamic encircling of the exceptional point for entangled states.a Schematic of the quantum walk evolution operator.b Riemann energy surface for evolution operator   i U i N  with  and 1  , other parameters are 0.2, 0 k   and 2 16    .
eigenstates of the evolution operator i U .The details of these eigenvalues and eigenstates are provided in Section 2 of Methods.

can exhibit behavior surrounding an EP by appropriately selecting the parameters 1  1  1  4  1  and 2  1  and 2  1 
and  .When the parameter asterisk in Fig.1bis chosen as the starting point, the total number of step) at the nth step, the variation of parameters constitutes a loop (black Loop 1) as shown in Fig.1c.The positive sign in the above equation corresponds to the entangled state evolving along the counter-clockwise path, while the negative sign corresponds to the clockwise path.Next, four Bell states the input to the system, and the evolutions are studied.To make the state evolution approximately adiabatically, the number of total steps along Loop 1 and  change slowly.The theoretical density matrices of the four output states are shown in Fig.2.Figs.2e-2h correspond to the case where and  change clockwise, while Figs.2i-2l correspond to the change counter-clockwise.For comparison, Figs.2a-2d show the density matrices of the input states.The white, yellow, red and blue asterisks in Figs.2a-2d corresponds to those labeled in Fig. 1b, which represent the , respectively.As shown in Figs.2e and 2f, when the input states encircle the EP clockwise (CW), the evolved output results are both very close to the entangled state 2  .The calculated fidelities are as high as 98.3% and 96.4%, respectively.For comparison, if the input states encircle the EP counter-clockwise (CCW), the output results are both very close to the entangled state with very high fidelities, see Figs. 2i and 2j.It indicates that encircling the EP enables asymmetric conversion between the entangled states 1

Figure 2 .
Figure 2. Theoretical results for Loop 1 encircling the EP.a-d Density matrices of the input It is found that these fidelity values are all above 0.97, indicating the forms are very close.If the parameters are tuned to make the eigenstates ( 1, 2,3, 4) j j   ideal Bell states, the output states will also be ideal Bell states.In addition, to achieve the above chiral switch, the evolution path of parameters cannot be far away from the EP.This chiral switch disappears if the evolution path of parameters are far away from the EP.For example, when the parameter values at the nth step are taken as: a path not enclosing the EP but away from it, which is shown as the red Loop 2 in Fig. 1c.Our results of the Bell state conversion show that the chiral behavior disappears, which the detailed results has been provided in S2 of Supplementary Materials.

states 1 , 2 , 3 , 4  1 C
as the initial states.Since the operator 1  acting on the four Bell states 1,2,3,4  can yield product states that are easy to prepare accurately, we directly prepare the states angles of half-wave plates (HWP) and quarter-wave plates (QWPs), before sending them into the multi-step QW.

Figure 3 .
Figure 3. Experimental setup.a State preparation.b Implementation of the encircling evolution around the exceptional point.c Measurement.

1 G
is very close to the identity matrix for relatively large N, i.e., 1 1 ii C C I    , where the detailed analysis is provided in Section 3 of Methods.In the experiment, one photon propagates in free space while the other photon enters the QW n M .The operators in n M can all be implemented experimentally.The rotation operator () R  is implemented using a combination of a green HWP at 0° and a black HWP at  .Two QWPs and one HWP together implement the conditional phase shift operator S .For the gain-loss operator G equating small (large) loss to gain (loss), where 1 0 l  and 2 1 l  .Similarly,  can be implemented by the equivalent gain-loss operator 1 L  .In the experiment, partially polarizing beam splitters (PPBS) are used to implement the operator L , while other loss operators-1  L are implemented using a sandwich-type HWP-PPBS A-HWP optical device.Moreover, to implement the symmetry-breaking operator     , a combination of two QWPs and one HWP is placed at the end of each step.For different n M , by changing the parameters 1

NC
, we decompose it into the product of a SWAP gate, controlled-not (CNOT) gate, and the operator T. The SWAP gate can exchange the states of two quantum bits.In experiments, it can be implemented by exchanging the upper and lower photons using mirrors.The CNOT gate is implemented by Hong-Ou-Mandel interference using a combination of two PPBS B and one PPBS C. For the operator T, different combinations of HWPs, QWPs, and PPBS D are placed in the upper and lower paths to implement it.The detailed experimental implementation is provided in Section 4 of Methods.

Figure 4 .
Figure 4. Experimental results of the chiral entanglement switching with encircling an EP.

8 N 1  and 2  2  1  1  and 2  3  and 4  , both change leads to 3  4 
Figs.4i and 4j, when circling the EP counterclockwise, the final entangled states obtained

Figure 5 .
Figure 5. Experimental results of fidelities.In a and b, under the perturbation of disorder, the fidelity between the output states and the ideal entangled states for different input states.The label "CW" denotes the path circling the EP clockwise in the experiment, and "CCW" means

4 .
Ten groups of perturbations are chosen, and the average results over these groups are shown in Fig.5.The blue bars represent the fidelities between the output states and the ideal entangled states without disorder, and the gray bars represent the fidelities with disorder for comparison.Fig.5ashows the cases of clockwise encircling of the EP.It can be seen that for the four different input Bell states, the fidelities between the output states and the ideal Bell states with disorder (gray bars) do not change much compared with the corresponding cases without disorder (blue bars), remaining at high values (above 0.85).Similar results are found for the cases of counterclockwise encircling of the EP in Fig.5b.The fidelities with disorder also do not change much compared to the case without disorder.This means the chiral switching of the entangled states does indeed exhibit robustness against disorder in the path parameters.Discussion and conclusion.The usual approach to achieving conversion of entanglement states is to precisely manipulate a two-qubit gate, and the conversion between different entangled states requires constructing different quantum gates.However, such an operation does not have topological protection characteristics, which is easily affected by environment and appears errors.In this work, we have provided effective scheme to realize robust operation of quantum entanglement states with high fidelity by designing quadruple degeneracy EPs.Because the designed Riemann energy surfaces with degeneracy EPs have the same eigenstates as the entangled states, asymmetric conversion between the entangled states can be realized by encircling the EP.Such manipulation for the entangled states is topologically protected due to the topological properties of the Riemann surface structure.Furthermore, the phenomena have been experimentally demonstrated by constructing the quantum walk platforms.The above discussions focus on the case for encircling the EPs.Recent investigations have shown that chiral state transfers can appear without encircling the EP or near EP 27-28 .In fact, our designed topologically protected entanglement switching can also work without encircling the EP or near EP.The detailed discussions have been given in S6 of Supplementary Materials.
in the theoretical design.In this way, the evolution operator at the nth step is

U
are the same, which means the same Riemann energy surfaces.

4 
are all larger than 97%.Therefore, the eigenstates of system with as Bell states.In this way, the efficient quantum control among bell states can be realized by encircling the EP.

L
partial polarization beam splitter (PPBS), an optical device with different transmittance and vertical polarizations of the incident light.The horizontal polarization in the experiment is set to be fully transmitted   can be achieved by a PPBS with another type of transmittance 

nM
in the optical platform, then the are given to complete the evolution encircling the EP.

.
is found that the difference between the operator N C and T SWAP CNOT  the coefficient -1 in the second row of corresponding matrices.Such difference can be eliminated by the phase plate.The operator T can be obtained with 1 T and 2 T .Since the operator 1 be implemented by the sandwich type HWP-PPBS D-HWP in the experiment.
The Jones matrices of the HWP and QWP are